Lemma 4.20.
Any quotient map of logoi is an epimorphism in the category of logoi.
Proof
Let \(\phi^*\colon T \to S\) be a quotient map and put \(K=\ker(\phi^*)\). The induced morphism \(T/K\to S\) is an equivalence of logoi. By the universal property of the quotient from Proposition 4.18, for every logos \(T'\) the restriction functor
\[(\phi^*)^*\colon \Hom_{\Logos}(S,T') = \Fun^{\lex,\colim}(S,T')^{\simeq} \hookrightarrow \Fun^{\lex,\colim}(T,T')^{\simeq} = \Hom_{\Logos}(T,T').\]
is fully faithful. This is precisely the assertion that \(\phi^*\) is an epimorphism in \(\Logos\).